An analytic solution to the Busemann-Petty problem on sections of convex bodies

Annals of Mathematics

Volumn 149, Issue 2, 1999, Pages 691-703

  Gardner, R J ^a   Koldobsky, A ^b   Schlumprecht, T ^c  

^a WESTERN WASHINGTON UNIVERSITY   (United States)

^b UNIVERSITY OF TEXAS AT SAN ANTONIO   (United States)

^c TEXAS A AND M UNIVERSITY   (United States)

Author keywords

Busemann Petty problem; Convex body; Fourier transform; Intersection body; Radon transform; Star body

Indexed keywords

EID: 0033435894     PISSN: 0003486X     EISSN: None     Source Type: Journal    
DOI: 10.2307/120978     Document Type: Article

References (30)

1. K. BALL, Some remarks on the geometry of convex sets, in: Geometric Aspects of Functional Analysis, (J. Lindenstrauss and V. D. Milman, eds.), Lecture Notes in Math. 1317, Springer-Verlag, New York (1988), 224-231.
2. J. BOURGAIN, On the Busemann-Petty problem for perturbations of the ball, Geom. Funct. Anal. 1 (1991), 1-13.
3. H. BUSEMANN and C. M. PETTY, Problems on convex bodies, Math. Scand. 4 (1956), 88-94.
4. H. FALLERT, P. GOODEY, and W. WEIl, Spherical projections and centrally symmetric sets, Adv. in Math. 129 (1997), 301-322.
5. R. J. GARDNER, Intersection bodies and the Busemann-Petty problem, Trans. A.M.S. 342 (1994), 435-445.
6. R. J. GARDNER, A positive answer to the Busemann-Petty problem in three dimensions, Ann. of Math. 140 (1994), 435-447.
7. Geometric Tomography, Cambridge University Press, Cambridge, 1995.
8. R. J. GARDNER, A. KOLDOBSKY, and T. SCHLUMPRECHT, An analytic solution to the Busemann-Petty problem, C. R. Acad. Sci. Paris, Sér. I Math. 328 (1999), 29-34.
9. I. M. GELFAND and G. E. SHILOV, Generalized Functions, Vol. 1. Properties and Operations, Academic Press, New York, 1964.
10. I. M. GELFAND and N. YA. VILENKIN, Generalized Functions, Vol. 4. Some Applications of Harmonic Analysis, Academic Press, New York, 1964.
11. A. GIANNOPOULOS, A note on a problem of H. Busemann and C. M. Petty concerning sections of symmetric convex bodies, Mathematika 37 (1990), 239-244.
12. P. GOODEY, E. LUTWAK, and W. WEIL, Functional analytic characterization of classes of convex bodies, Math. Z. 222 (1996), 363-381.
13. P. GOODEY and W. WEIL, Intersection bodies and ellipsoids, Mathematika 42 (1995), 295-304.
14. S. HELGASON, The Radon Transform, Prog. in Math. 5, Birkhäuser, Boston, 1980.
15. A. KOLDOBSKY, Schoenberg's problem on positive definite functions, Alg. and Anal. 3 (1991), 78-85. English translation in St. Petersburg Math. J. 3 (1992), 563-570.
16. An application of the Fourier transform to sections of star bodies, Israel J. Math. 106 (1998), 157-164.
17. Intersection bodies, positive definite distributions and the Busemann-Petty problem, Amer. J. Math. 120 (1998), 827-840.
18. 4, Adv. in Math. 136 (1998), 1-14.
19. Second derivative test for intersection bodies, Adv. in Math. 136 (1998), 15-25.
20. D. G. LARMAN and C. A. ROGERS, The existence of a centrally symmetric convex body with central sections that are unexpectedly small, Mathematika 22 (1975), 164-175.
21. P. LÈVY, Théorie de l'Addition de Variable Aléatoires, Gauthier-Villars, Paris (1937),
22. E. LUTWAK, Intersection bodies and dual mixed volumes, Adv. in Math. 71 (1988), 232-261.
23. M. MEYER, On a problem of Busemann and Petty, unpublished manuscript.
24. V. D. MILMAN and A. PAJOR, Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space, in Geometric Aspects of Functional Analysis (J. Lindenstrauss and V. D. Milman, eds.), Lecture Notes in Math. 1376, Springer-Verlag, New York (1989), 64-104.
25. n, Mathematika 39 (1992), 258-266.
26. R. SCHNEIDER, Convex Bodies: The Brunn-Minkowski Theory, Cambridge University Press, Cambridge, 1993.
27. GAOYONG ZHANG, Centered bodies and dual mixed volumes, Trans. A.M.S. 345 (1994), 777-801.
28. 4, Ann. of Math. 140 (1994), 331-346.
29. 4, Ann. of Math. 149 (1999), 535-543.
30. Intersection bodies and polytopes, Mathematika, to appear.